# Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theorems on the Solution of Euler’s Differential Equation

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

**Example**

**1.**

**Remark**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**2.**

**Theorem**

**3.**

## 3. Preliminaries on Distribution Theory

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

#### 3.1. Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Condition**

**1.**

**Lemma**

**8.**

**Proof.**

#### 3.2. Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$, as well as in the Space ${\mathcal{D}}^{\prime}$

**Remark**

**4.**

**Remark**

**5.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

#### 3.3. Regular Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Lemma**

**14.**

**Lemma**

**15.**

## 4. Euler’s Equation in the Space of Distributions ${\mathcal{D}}_{R}^{\prime}$

**Remark**

**6.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Remark**

**7.**

**Lemma**

**18.**

**Proof.**

**Theorem**

**4.**

**Remark**

**8.**

**Remark**

**9.**

**Proof**

**of**

**Theorem**

**4.**

**Example**

**3.**

**Remark**

**10.**

**Example**

**4.**

**Theorem**

**5.**

**Proof.**

**Remark**

**11.**

**Example**

**5.**

**Theorem**

**6.**

**Remark**

**12.**

**Example**

**6.**

**Example**

**7.**

## 5. Euler’s Equation Studied in Nonstandard Analysis

**Theorem**

**7.**

**Proof.**

#### 5.1. AC-Laplace Transform of Euler’s Equation

**Theorem**

**8.**

**Remark**

**13.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Solution of a Linear Differential Equation with Constant Coefficients

**Theorem**

**A1.**

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Morita, T.; Sato, K.-i. Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus. *Mathematics* **2020**, *8*, 2117.
https://doi.org/10.3390/math8122117

**AMA Style**

Morita T, Sato K-i. Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus. *Mathematics*. 2020; 8(12):2117.
https://doi.org/10.3390/math8122117

**Chicago/Turabian Style**

Morita, Tohru, and Ken-ichi Sato. 2020. "Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus" *Mathematics* 8, no. 12: 2117.
https://doi.org/10.3390/math8122117